You've probably heard of a curve called the parabola, and you probably interpret this as meaning it's a function something like $y = x^2$. However there is another way to define the parabola. If you draw a line (called the directrix) and then choose a point (called the focus) not on that line then the set of points that are an equal distance from the directrix and the focus form a parabola.
(the picture is from this web site)
All conic sections have a focus. For the circle the focus is the centre, and you may have heard that the planets orbit in ellipses with the Sun at one focus. Anyhow, with some effort you can prove that any light ray from the focus reflects off the parabola parallel to the axis of symmetry, or any ray parallel to the axis of symmetry is reflected through the focus. This property is the basis of parabolic reflectors.
But what has this to do with spherical mirrors? Well, as long as you keep the curvature of a spherical mirror small it is very similar to a parabola and it shares the property of reflecting parallel rays onto (almost) a single point. Hence statement (1) in your question.
However because a spherical mirror isn't the same as a parabolic mirror the approximation breaks down as the spherical mirror gets bigger. Spherical mirrors actually focus parallel rays onto a surface called a caustic, not onto a single point.